In western Kansas, the summer density of hailstorms is estimated at about 2.6 storms per 5 square miles. In most cases, a hailstorm damages only a relatively small area in a square mile. A crop insurance company has insured a tract of 8 square miles of Kansas wheat land against hail damage. Let r be a random variable that represents the number of hailstorms this summer in the 8-square-mile tract.

(a) Explain why a Poisson probability distribution is appropriate for r.

Hail storms in western Kansas are a common occurrence. It is reasonable to assume the events are dependent. Hail storms in western Kansas are a rare occurrence. It is reasonable to assume the events are dependent. Hail storms in western Kansas are a common occurrence. It is reasonable to assume the events are independent. Hail storms in western Kansas are a rare occurrence. It is reasonable to assume the events are independent.

What is λ for the 8-square-mile tract of land? Round λ to the nearest tenth so that you can use Table 4 of Appendix II for Poisson probabilities.

(b) If there already have been two hailstorms this summer, what is the probability that there will be a total of four or more hailstorms in this tract of land? Compute P(r≥ 4 | r ≥ 2). (Use 4 decimal places.)

(c) If there already have been three hailstorms this summer, what is the probability that there will be a total of fewer than six hailstorms? Compute P(r < 6 | r ≥ 3). (Use 4 decimal places.)